Some Researches On Iterative Methods For Solving Saddle Point Problems

Saddle point problems have many practical applications in scientific comput-ing and engineering fields,such as computing fluid dynamics,electronic networks,elliptic partial differential equations of mixed finite element approximation,image processing and so on,so the fast and effective method for solving the saddle point problems is a hot issue in the study of mathematics.The main contents of this thesis are:In Introduction,the development and research status of saddle point problems are summarized,and some basic definitions and notations are introduced.In Chapter 1,the PAHSS-PTS splitting iteration method for solving the non-singular saddle point problems is studied.The new method is based on PAHSS iteration method and preconditioned triangular splitting technique and it is conver-gent under certain conditions.In Chapter 2,the IU-Uzawa iteration method for solving singular saddle point problems is studied.The shift splitting technique is applied to the first block of the coefficient matrix of the singular saddle point problems,and then the new method is obtained by combining the Uzawa algorithm.The semi-convergence of the IU-Uzawa method is proved,and the numerical experiments show that the new method is effective.In Chapter 3,we generalized the ULT splitting iteration method and presented the parameterized ULT splitting iteration method for the singular and nonsingular saddle point problems.The convergence and the semi-convergence of this method are verified.In Chapter 4,a local HSS triangular splitting iteration method for solving non-symmetric saddle point problems is studied.The HSS splitting technique is applied to the first block of the coefficient matrix of the nonsymmetric saddle point prob-lems,and then the LHSST iteration method is obtained by combining the triangular splitting.In Chapter 5,the research work of this thesis is summarized,and put forward the idea of future research work and the problem to be solved.